Probability distributions

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A Probability Distribution is a special kind of distribution and Joe Schmuller demonstrates how very easy it is to assign a probability to a coin toss or rolling of a die. Probabilty distributions assigns a probability to every possible outcome of an experiment. Joe reminds you to think of each possible outcome as the value of a random variable.

- [Instructor] Let's look at probability distributions.A probability distribution is aspecial kind of distribution.It assigns a probability to everypossible outcome of an experiment.Think of each outcome as a possiblevalue of a random variable.A probability distribution can be discrete,and so its random variable is a setof possible outcomes that you can count.It's very easy to assign a probabilityto each one like a coin toss or rolling a die.

And here's a simple example.Tossing a coin.The random variable is coin tossbut we can refer to it as X.The possible values are head and tail.We can assign arbitrarily head equals oneand tail equals zero.The possible values of the random variableX are one and zero.And here's a picture of the probability distributionfor tossing a fair coin.Each possible value, one or zero,has an equal probability of occurrence,and that probability is one half.

This type of probability distributionis called a probability mass function.A probability distribution can be continuous.That means its random variableis on a continuum.The possible outcomes are not countable.The random variable can take on any valuebetween two specified values.Here's an example of what I mean.When we measure a person's height,the ruler's precision limits the accuracy,so probability is assigned to an interval,not to an exact number.

For example, instead of the probabilitythat a person is 69 inches tall,we'd be concerned with the probabilitythat their height is between68 inches and 70 inches.These kinds of distributions look like this.Possible values are on a continuum,the number of outcomes is uncountable.This type of probability distributionis called a probability density function.Note that probability density is on the Y axis.Probability density is a math conceptthat enables us to use area under the curveas probability.

A probability density function isoften based on a complex equation.Every distribution has a mean and a variance,and a probability distribution is no exception.Calculating the mean and varianceis easier for a discrete distributionthan for a continuous distribution.For a continuous distribution,we'd have to get into some sophisticated mathematicsand we won't do that.The mean of a discrete probability distributionis also called the expected value.To calculate the expected value,you multiply each outcome times its probability,add the products,and the result is the expected value.

Applying all this to tossing a coin,the outcomes are zero or one,each one has a probability equal to .5,so the expected value is(0)(.5) + (1)(.5)which comes out to .5.To calculate the variance of adiscrete probability distribution,you subtract the expected value from each outcomeand square the differences.Multiply each square difference byits corresponding probability,and the sum of the results is the variance,also labeled as V(x).

It's square root is the standard deviation.Now applying this to tossing a coin,the variance, (0-.5) squared times .5,plus (1-.5) squared times .5,comes out to .25,and the standard deviation is thesquare root of .25, which is .5.And now, the mean and the variancefor rolling a die.Expected value, as you can see, works out to 3.5,and the variance, is 2.92.

The standard deviation is thesquare root of 2.92, or 1.71.In summation, we talked aboutprobability distributions and howthey can be discrete or continuous,and we showed how to calculate themean and the variance of a discrete distribution.

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Author

Updated

6/2/2016

Released

12/23/2015

Understanding statistics is more important than ever. Statistical operations are the basis for decision making in fields from business to academia. However, many statistics courses are taught in cookbook fashion, with an emphasis on a bewildering array of tests, techniques, and software applications. In this course, part one of a series, Joseph Schmuller teaches the fundamental concepts of descriptive and inferential statistics and shows you how to apply them using Microsoft Excel.

He explains how to organize and present data and how to draw conclusions from data using Excel's functions, calculations, and charts, as well as the free and powerful Excel Analysis ToolPak. The objective is for the learner to fully understand and apply statistical concepts—not to just blindly use a specific statistical test for a particular type of data set. Joseph uses Excel as a teaching tool to illustrate the concepts and increase understanding, but all you need is a basic understanding of algebra to follow along.